Exact and efficient computations on circles in CGAL

Exact and efficient computations on circles in C

- Experiments

Pedro M. M. de Castro, Sylvain Pion, Monique Teillaud

European Workshop on Computational Geometry Graz - 2007

The Computational Geometry Algorithms Library Open Source project

www.cgal.org

>400.000 lines ofC++code

>3.000 pages manual

∼12.000 downloads per year

∼850 users on public mailing list

∼50 developers licenses LGPL or QPL start-up GeometryFactory interfaces: Python,Scilab

Robustness and efficiency Quality:

Editorial board (10 members)

Test-suites each night ...

kernels

Kernel=

elementary geometric objects (points, segments,. . . )

elementary operations on them

(intersection tests, intersection computations,. . . ) Up to release 3.1 (Dec’04):

essentiallylinearobjects Release 3.2 (May ’06):

2Dcircularkernel

[P.-T.]

2D circular kernel design

[Emiris-Kakargias-P.-T.-Tsigaridas socg’04]

Guidelines:

code reuse:

ability toreuse the CGALkernelfor points, circles, number types,. . .

flexibility:

possibility touse other implementationsfor points, circles, number types,. . .

possibility touse several algebraic implementations

template < LinearKernel, AlgebraicKernel > class Circular_kernel

2D circular kernel design

[Emiris-Kakargias-P.-T.-Tsigaridas socg’04]

Guidelines:

code reuse:

ability toreuse the CGALkernelfor points, circles, number types,. . .

flexibility:

possibility touse other implementationsfor points, circles, number types,. . .

possibility touse several algebraic implementations

template < LinearKernel, AlgebraicKernel >

class Circular_kernel

2D circular kernel design

[Emiris-Kakargias-P.-T.-Tsigaridas socg’04]

template < LinearKernel, AlgebraicKernel >

class Circular_kernel

Types

Must be defined byLinear_kernel

basic number types, points, lines,. . . Must be defined byAlgebraic_kernel

algebraic numbers, polynomials Defined byCircular_kernel

Circular_arc_2,Circular_arc_point_2

Predicates

e.g. intersection tests, comparisons of intersection points,. . . exactnessiscrucialfor geometric algorithms Constructions

e.g. computation of intersection points

2D circular kernel design

[Emiris-Kakargias-P.-T.-Tsigaridas socg’04]

template < LinearKernel, AlgebraicKernel >

class Circular_kernel

Types

Must be defined byLinear_kernel

basic number types, points, lines,. . . Must be defined byAlgebraic_kernel

algebraic numbers, polynomials Defined byCircular_kernel

Circular_arc_2,Circular_arc_point_2 Predicates

e.g. intersection tests, comparisons of intersection points,. . . exactnessiscrucialfor geometric algorithms Constructions

e.g. computation of intersection points

Representation

CGALCircle_2:

center

squared radius (rational)

Circular_arc_2:

supporting circleCircle_2

2Circular_arc_point_2 (algebraic)

Circular_arc_point_2 root of system

(system = 2 equations of circles) (algebraic)

Representation

CGALCircle_2:

center

squared radius (rational)

Circular_arc_2:

supporting circleCircle_2

2Circular_arc_point_2 (algebraic)

Circular_arc_point_2 root of system

(system = 2 equations of circles) (algebraic)

Experiments

Computation of arrangements with CGAL3.2 package [Wein-Fogel-Zukerman-Halperin]

* onreal industrial dataofVLSImodels

vlsi_7

VLSI N V E F

vlsi_1 11,929 20,649 26,468 6,385

vlsi_2 9,663 8,556 9,439 887

vlsi_3 35,449 101,758 163,316 61,887

vlsi_4 12,937 81,096 143,049 61,986

vlsi_5 4,063 40,636 77,598 36,965

vlsi_6 74,052 547,250 1,016,460 470,480

vlsi_7 89,918 495,209 878,799 383,871

vlsi_8 123,175 370,304 555,412 190,031

vlsi_9 139,908 1,433,248 2,690,530 1,257,684

* andsynthetic models(very dense and sparse)

Experiments - Results

Pentium 4, 2.5 GHz, 1GB, Linux (2.4.20 Kernel) g++4.0.2, -DNDEBUG -O2 times in seconds Input CGAL3.2 SpecificCGAL CGAL3.3

before arrangements after

vlsi_1 28.0 8.55 4.61

vlsi_2 48.0 2.59 1.31

vlsi_3 135 26.7 21.8

vlsi_4 569 26.9 25.4

vlsi_5 125 14.3 14.8

vlsi_6 611 137 134

vlsi_7 690 192 169

vlsi_8 3,650 220 136

vlsi_9 2,320 581 492

Very dense 335 77.9 76.2

Sparse 0.91 0.51 0.21

Caching

bit-field(2 bytes) inCircular_arc_2 Stores information like

monotone arc full circle . . .

caching intersections

ofeach pairof supporting circles

memory overflow on big data sets−→abandoned

First enhancements of algebraic number type

Optimizingparticular cases

cases when algebraic numbers are in fact rational (intersection of tangent circles)

Arithmetic filtering

general idea for comparison

compute approximate values + error boundε ifεsmall enough: conclude

otherwise compute exact values

Reference counting

High cost of copying coefficients of polynomials

(multiprecision numbers) Storehandlesto objects instead of objects

Copying = copying the reference

Intermediate results

(some of the enhancements were already in the CGALspecific implementation for arrangements)

Input before now SpecificCGAL

arrangements

vlsi_1 28.0 8.42 8.55

vlsi_2 48.0 3.01 2.59

vlsi_3 135 25.8 26.7

vlsi_4 569 33.4 26.9

vlsi_5 125 15.2 14.3

vlsi_6 611 135 137

vlsi_7 690 185 192

vlsi_8 3,650 445 220

vlsi_9 2,320 525 581

Very dense 335 84.5 77.9

Sparse 0.91 0.40 0.51

Further improvements

Representation ofalgebraic numbers

Initialrepresentation ofalgebraic numbers:

root ofax²+bx +c 3rationalcoefficientsa,b,c

(multi-precision-GMP or CGAL::MP_Float) one boolean

Computation of approximation and error for filtering:

(−b±√

b²−4ac)/2a . . .

Newrepresentation ofalgebraic numbers:

α+β√ γ 3 numbersα, β, γ

allows less efficient comparisons, but

reduces the lengthsof multi-precision numbers (and allows additions in easy cases)

Further improvements

Representation ofalgebraic numbers

Initialrepresentation ofalgebraic numbers:

root ofax²+bx +c 3rationalcoefficientsa,b,c

(multi-precision-GMP or CGAL::MP_Float) one boolean

Computation of approximation and error for filtering:

(−b±√

b²−4ac)/2a . . .

Newrepresentation ofalgebraic numbers:

α+β√ γ 3 numbersα, β, γ

allows less efficient comparisons, but

reduces the lengthsof multi-precision numbers (and allows additions in easy cases)

Further improvements

Representation ofalgebraic numbers

Histograms (on vlsi_8)

quantity of numbers vs. lengths (number of digits) Initial

New

Further improvements

Geometric filtering

Forpredicates(e.g. intersection tests) usebounding boxof objects:

Test intersection on the boxes if they don’t intersect: conclude

otherwise test intersection on the exact objects

Results

Input CGAL3.2 SpecificCGAL CGAL3.3 before arrangements after

vlsi_1 28.0 8.55 4.61

vlsi_2 48.0 2.59 1.31

vlsi_3 135 26.7 21.8

vlsi_4 569 26.9 25.4

vlsi_5 125 14.3 14.8

vlsi_6 611 137 134

vlsi_7 690 192 169

vlsi_8 3,650 220 136

vlsi_9 2,320 581 492

Very dense 335 77.9 76.2

Sparse 0.91 0.51 0.21

Conclusion and future work

Combination of

generalityandre-usabilityof the functionality robustnessandefficiency

Integration in CGAL3.3.

Future work

manipulations of spheres, circles, and circular arcs in3D

(submitted to CGAL) [dC.-T.]

filtering ofconstructions?

[Funke-Mehlhorn’00, Fabri-P. lcsd’06]

Work partially supported by the EU STREP Project IST-006413